Geodesic
Chern slopes of simply connected complex surfaces of general type are dense in [2,3]
Xavier Roulleau1 ; Giancarlo Urzúa2
1 Unité de Formation Mathématiques, Université de Poitiers, Poitiers, France
2 Pontificia Universidad, Católica de Chile, Santiago, Chile
Annals of mathematics, Tome 182 (2015) no. 1, pp. 287-306.

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We prove that for any number $r \in [2,3]$, there are spin (resp. nonspin and minimal) simply connected complex surfaces of general type $X$ with $c_1^2(X)/c_2(X)$ arbitrarily close to $r$. In particular, this shows the existence of simply connected surfaces of general type arbitrarily close to the Bogomolov-Miyaoka-Yau line. In addition, we prove that for any $r \in [1,3]$ and any integer $q\geq 0$, there are minimal complex surfaces of general type $X$ with $c_1^2(X)/c_2(X)$ arbitrarily close to $r$ and $\pi_1(X)$ isomorphic to the fundamental group of a compact Riemann surface of genus $q$. %A central ingredient is a new family of special arrangements of elliptic curves in the projective plane.
DOI : 10.4007/annals.2015.182.1.6

Xavier Roulleau 1 ; Giancarlo Urzúa 2

1 Unité de Formation Mathématiques, Université de Poitiers, Poitiers, France
2 Pontificia Universidad, Católica de Chile, Santiago, Chile 
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Xavier Roulleau; Giancarlo Urzúa. Chern slopes of simply connected complex surfaces of general type are dense in [2,3]. Annals of mathematics, Tome 182 (2015) no. 1, pp. 287-306. doi : 10.4007/annals.2015.182.1.6. http://geodesic.mathdoc.fr/articles/10.4007/annals.2015.182.1.6/

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